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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj132 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj132.1 | ⊢ (𝜑 ↔ ∃𝑥(𝜓 → 𝜒)) |
Ref | Expression |
---|---|
bnj132 | ⊢ (𝜑 ↔ (𝜓 → ∃𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj132.1 | . 2 ⊢ (𝜑 ↔ ∃𝑥(𝜓 → 𝜒)) | |
2 | 19.37v 1998 | . 2 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (𝜓 → ∃𝑥𝜒)) | |
3 | 1, 2 | bitri 274 | 1 ⊢ (𝜑 ↔ (𝜓 → ∃𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wex 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 |
This theorem depends on definitions: df-bi 206 df-ex 1786 |
This theorem is referenced by: bnj996 32915 |
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